Normal-Wishart distribution

Normal-Wishart
Notation
Parameters	location (vector of real); (real); scale matrix (pos. def.); (real)
Support	covariance matrix (pos. def.)
PDF

In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).^[1]

Definition

Suppose

{\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Lambda }}\sim {\mathcal {N}}({\boldsymbol {\mu }}_{0},(\lambda {\boldsymbol {\Lambda }})^{-1})

has a multivariate normal distribution with mean ${\boldsymbol {\mu }}_{0}$ and covariance matrix $(\lambda {\boldsymbol \Lambda })^{{-1}}$ , where

{\boldsymbol \Lambda }|{\mathbf {W}},\nu \sim {\mathcal {W}}({\boldsymbol \Lambda }|{\mathbf {W}},\nu )

has a Wishart distribution. Then $({\boldsymbol \mu },{\boldsymbol \Lambda })$ has a normal-Wishart distribution, denoted as

({\boldsymbol \mu },{\boldsymbol \Lambda })\sim {\mathrm {NW}}({\boldsymbol \mu }_{0},\lambda ,{\mathbf {W}},\nu ).

Characterization

Probability density function

f({\boldsymbol \mu },{\boldsymbol \Lambda }|{\boldsymbol \mu }_{0},\lambda ,{\mathbf {W}},\nu )={\mathcal {N}}({\boldsymbol \mu }|{\boldsymbol \mu }_{0},(\lambda {\boldsymbol \Lambda })^{{-1}})\ {\mathcal {W}}({\boldsymbol \Lambda }|{\mathbf {W}},\nu )

Properties

Marginal distributions

By construction, the marginal distribution over ${\boldsymbol {\Lambda }}$ is a Wishart distribution, and the conditional distribution over $\boldsymbol\mu$ given ${\boldsymbol {\Lambda }}$ is a multivariate normal distribution. The marginal distribution over $\boldsymbol\mu$ is a multivariate t-distribution.

Posterior distribution of the parameters

After making $n$ observations ${\boldsymbol {x}}_{1},\dots ,{\boldsymbol {x}}_{n}$ , the posterior distribution of the parameters is

({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{n},\lambda _{n},\mathbf {W} _{n},\nu _{n}),

where

\lambda _{n}=\lambda +n,

{\boldsymbol {\mu }}_{n}={\frac {\lambda {\boldsymbol {\mu }}_{0}+n{\boldsymbol {\bar {x}}}}{\lambda +n}},

\nu _{n}=\nu +n,

\mathbf {W} _{n}^{-1}=\mathbf {W} ^{-1}+\sum _{i=1}^{n}({\boldsymbol {x}}_{i}-{\boldsymbol {\bar {x}}})({\boldsymbol {x}}_{i}-{\boldsymbol {\bar {x}}})^{T}+{\frac {n\lambda }{n+\lambda }}({\boldsymbol {\bar {x}}}-{\boldsymbol {\mu }}_{0})({\boldsymbol {\bar {x}}}-{\boldsymbol {\mu }}_{0})^{T}.

^[2]

Generating normal-Wishart random variates

Generation of random variates is straightforward:

Sample ${\boldsymbol {\Lambda }}$ from a Wishart distribution with parameters $\mathbf {W}$ and $\nu$
Sample $\boldsymbol\mu$ from a multivariate normal distribution with mean ${\boldsymbol {\mu }}_{0}$ and variance $(\lambda {\boldsymbol \Lambda })^{{-1}}$

Related distributions

The normal-inverse Wishart distribution is essentially the same distribution parameterized by variance rather than precision.
The normal-gamma distribution is the one-dimensional equivalent.
The multivariate normal distribution and Wishart distribution are the component distributions out of which this distribution is made.

Notes

↑ Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media. Page 690.
↑ Cross Validated, https://stats.stackexchange.com/q/324925

References

Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[bishop-1] Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media. Page 690.

[2] Cross Validated, https://stats.stackexchange.com/q/324925

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

Notation	$({\boldsymbol \mu },{\boldsymbol \Lambda })\sim {\mathrm {NW}}({\boldsymbol \mu }_{0},\lambda ,{\mathbf {W}},\nu )$
Parameters	${\boldsymbol \mu }_{0}\in {\mathbb {R}}^{D}\,$ location (vector of real) $\lambda >0\,$ (real) ${\mathbf {W}}\in {\mathbb {R}}^{{D\times D}}$ scale matrix (pos. def.) $\nu >D-1\,$ (real)
Support	${\boldsymbol \mu }\in {\mathbb {R}}^{D};{\boldsymbol \Lambda }\in {\mathbb {R}}^{{D\times D}}$ covariance matrix (pos. def.)
PDF	$f({\boldsymbol \mu },{\boldsymbol \Lambda }\|{\boldsymbol \mu }_{0},\lambda ,{\mathbf {W}},\nu )={\mathcal {N}}({\boldsymbol \mu }\|{\boldsymbol \mu }_{0},(\lambda {\boldsymbol \Lambda })^{{-1}})\ {\mathcal {W}}({\boldsymbol \Lambda }\|{\mathbf {W}},\nu )$